A general approach to correlation and linear dependence between random vectors with regular or singular covariance matrix

Abstract

A direction-depending measure for linear dependences between multivariate random voctors is given: the Direction Correlation (DC). It measures the accuracy of the Best Linear approximation (Prediction) of a random vector by another in view of certain risk functional. The DC is a generalistion of the usual multiple correlation coefficient. The so called Partial DC – as the corresponding generalisation of the parial miltiple correlation coefficient – can proved to be the DC of two random vectors after eliminating the linear influences of a third one. The Canonical Correlations are special maximal DC's and monoton in additional approximation variates (predictors) also for singular covariance matrices.